Abstract
The paper develops a general scheme for viscoelastic materials, where the constitutive properties are described by means of measures of strain, stress, heat flux, and their time derivatives. The constitutive functions are required to be consistent with the second law of thermodynamics. Indeed, a new view is associated with the second law: the non-negative expression of the entropy production is set equal to a further constitutive function. The introduction of the entropy production as a constitutive function allows for a much wider range of models. Within this range, a scheme to obtain nonlinear models of thermo-viscoelastic materials subject to large deformations is established. Notably, the Kelvin–Voigt, Maxwell, Burgers, and Oldroyd-B viscoelastic models, along with the Maxwell–Cattaneo heat conduction, are obtained as special cases. The scheme allows also for modelling the visco-plastic materials, such as the Prandtl–Reuss work-hardening function and the Bingham–Norton fluid.
Highlights
Viscoelasticity, especially the model of linear viscoelastic solids, traces back to Boltzmann [1], who, in essence, considered an elastic material with memory
While sometimes the model is motivated by rheological analogs [34] and framed within a scheme with intermediate reference configurations [35], here, the model is investigated within the thermodynamic approach developed so far
Consistent viscoelastic models are established on the basis of the following points
Summary
Viscoelasticity, especially the model of linear viscoelastic solids, traces back to Boltzmann [1], who, in essence, considered an elastic material with memory. The purpose of this paper is to establish thermodynamically consistent models of viscoelastic materials characterized by rate (or differential) equations for stress and strain. The pertinent variables of the constitutive equations comprise stress, strain, heat flux, and their time derivative It is a key point of our approach that the entropy production enters as a non-negative constitutive function. The main advantage of the present approach is the possibility of establishing nonlinear, thermodynamically consistent, objective rate-type relations describing memory and dissipative effects. It is worth mentioning References [10,11], where viscoelastic models are developed in the relativistic Landau–Lifshitz frame on the basis of Onsager’s linear non-equilibrium thermodynamics. Our scheme involves properties and processes such as nonlinearities and hysteretic effects [8,12,13]
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